Logarithmic potential appears as a solution of the two-dimensional Poisson equation, and thus, describes the interaction in a two-dimensional Coulomb gas, so there is no mystery in the appearance of elliptic and Jacobi theta functions in the vortices-driven Berezinskii-Kosterlitz-Thouless transition [1]}, [2]}, [3]}, [4]}.
Indeed, applying the theory of conformal mappings which hold at the critical point of second order phase transitions in two-dimensional systems, these special functions enable us to obtain a closed expression for the correlation functions in \(XY\)  models [5]}. More generically, they appear in the Schwarz-Christoffel mapping and related conformal mappings in the complex plane [6]}, [7]}, [8]}, [9]}.
